ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

doi:10.1007/s10473-021-0115-3

数学物理学报(英文版) ›› 2021, Vol. 41 ›› Issue (1): 257-282.doi: 10.1007/s10473-021-0115-3

ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

陈化¹, 刘念²

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. School of Science, Wuhan University of Technology, Wuhan 430070, China

收稿日期:2019-10-09 修回日期:2020-06-07 出版日期:2021-02-25 发布日期:2021-04-06
通讯作者: Nian LIU E-mail:nianliu@whut.edu.cn
作者简介:Hua CHEN,E-mail:chenhua@whu.edu.cn
基金资助:
This work was supported by NFSC (11871387 and 11931012).

ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

Hua CHEN¹, Nian LIU²

1. School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China;
2. School of Science, Wuhan University of Technology, Wuhan 430070, China

Received:2019-10-09 Revised:2020-06-07 Online:2021-02-25 Published:2021-04-06
Contact: Nian LIU E-mail:nianliu@whut.edu.cn
About author:Hua CHEN,E-mail:chenhua@whu.edu.cn
Supported by:
This work was supported by NFSC (11871387 and 11931012).

摘要/Abstract

摘要： In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted $p$-Sobolev spaces and the weighted corner type Sobolev inequality, the Poincar$\acute{e}$ inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family.

关键词: coupled parabolic equations, totally characteristic degeneracy, singular potentials, asymptotic stability, blow-up

Abstract: In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted $p$-Sobolev spaces and the weighted corner type Sobolev inequality, the Poincar$\acute{e}$ inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family.

Key words: coupled parabolic equations, totally characteristic degeneracy, singular potentials, asymptotic stability, blow-up

中图分类号:

35K65

陈化, 刘念. ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS[J]. 数学物理学报(英文版), 2021, 41(1): 257-282.

Hua CHEN, Nian LIU. ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS[J]. Acta mathematica scientia,Series B, 2021, 41(1): 257-282.

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ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

ON THE EXISTENCE WITH EXPONENTIAL DECAY AND THE BLOW-UP OF SOLUTIONS FOR COUPLED SYSTEMS OF SEMI-LINEAR CORNER-DEGENERATE PARABOLIC EQUATIONS WITH SINGULAR POTENTIALS

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